Almost Exact Exchange At Almost No Cost

file: ECPG14˙KPP
Total pages: 6
Almost Exact Exchange At Almost No Cost
Peter Elliott,1 Attila Cangi,1 Stefano Pittalis,2 E.K.U. Gross,1 and Kieron Burke3
1
Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle (Saale), Germany
2
CNR-Istituto di Nanoscienze, Via Campi 213A, I-41125 Modena, Italy
3
Department of Chemistry, University of California, Irvine, CA 92697, USA
(Dated: August 19, 2014)
A recently developed semiclassical approximation to exchange in one dimension is shown to be
almost exact, with essentially no computational cost. The variational stability of this approximation
is tested, and its far greater accuracy relative to local density functional calculations demonstrated.
Even a fully orbital-free potential-functional calculation (no orbitals of any kind) yields little error
relative to exact exchange, for more than one orbital.
BurkeID: BG90001
0.08
Error (a.u.)
0.06
0.04
LDAX
scX*
scKX*
0.02
0
2
4
6
8
N
Error in the total energy made by LDA exchange (LDAX), semiclassical exchange (scX), and semiclassical kinetic and
exchange (scKX) for N spin-unpolarized, interacting fermions in a well as compared to an exact-exchange (EXX) calculation.
For both scX and scKX, the exchange energy was given by a recently developed potential functional theory approximation[22]
which consists of a simple formula for the exchange energy in terms of the KS potential.In this case we evaluated on the
self-consistent potential of a LDA exchange calculation. Clearly we are achieving an accuracy on par with EXX while our
formula for exchange only requires a fraction of the computational effort of EXX.
PACS numbers: 31.15.E-,71.15.Mb,31.15.xg
proximations that are continuously created[6].
Electronic structure problems in chemistry, physics,
and materials science are often solved via the KohnSham method of density functional theory (DFT)[1, 2],
which balances accuracy with computational cost. For
any practical calculation, the exchange-correlation (XC)
energy must be approximated as a functional of the density. The basic theorems of DFT guarantee its uniqueness, but give no hint about constructing approximations. The early local density approximation (LDA)[2],
much used in solid state physics, was the starting point
for today’s more accurate methods such as the generalized gradient[3, 4] and hybrid[5] approximations. But a
systematic approach for deriving these has not yet been
found, a fact that is reflected by the plethora of XC ap-
This lack also inspires many approaches beyond traditional DFT, such as orbital-dependent functionals like
exact exchange (EXX) [7, 8], use of the random phase
approximation[9], and (first-order) density matrix functional theory[10]. While any of these can produce
higher accuracy, their computational cost is typically
much greater, and none have yet yielded a universal
improvement over existing Kohn-Sham (KS) DFT. Perhaps the most ubiquitous DFT method is that of hybrid functionals, which replace some generalized gradient
exchange with exact exchange. Hybrids are now standard in molecular calculations, and yield more accurate
thermochemistry in most cases[6]. Furthermore, range1
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ties, (ii) give an algorithm for solving this equation, (iii)
implement that algorithm in 1d, and (iv) perform purely
PFT calculations.
To begin, the ground-state energy of N electrons in an
external potential v(r) is given by
separated hybrids [11], where the exchange is treated in
a Hartree-Fock fashion, typically yield much improved
band gaps for many bulk solids[12]. However, their
computational cost in plane-wave codes can be up to
a thousand times higher than that of generalized gradient approximation calculations, making such methods
much less useful in practice[13]. Implicit within quantum mechanics is that exact or approximate solutions of
the Schrodinger equation are functionals of the potential, but the methodology of DFT is built around the
density instead. Pioneering work[14] derived the duality of density and potential functionals in the context
of an orbital-dependent KS-DFT calculation. More recently, the formalism of a pure potential functional theory (PFT) has been developed[15–17], and approximations for non-interacting fermions in simple model systems have been tested[18, 19]. The leading corrections
to Thomas-Fermi theory are explicit functionals of the
potential[18, 20, 21], and inclusion of these corrections
yields approximations that are typically much more accurate than their DFT counterparts.
E0 = minhΨ | Tˆ + Vˆee + Vˆ | Ψi ,
Ψ
F [v] = hΨ0 [v] | Tˆ + Vˆee | Ψ0 [v]i
v
˜
where n[v](r) is the ground-state density of v(r). In the
exact case, v˜(r) = v(r), but this is not necessarily true
for approximations.
In previous work[17], it was shown that in PFT,
once n[v](r) is given, F [v] can be deduced, either by
a coupling-constant integral or a virial relation. When
applied to non-interacting fermions, an approximation
nS [vS ](r) yields an approximation TS [vS ], where vS (r) is
the potential in this non-interacting case. Now we introduce a direct approximation to the XC energy, EXC [vS ], as
a functional of the KS potential, and ask: How can these
two approximations be used to find the ground-state energy of interacting fermions? This question differs from
that of deducing the KS equations in DFT, because here
the approximation is a potential functional, not a density
(or orbital-dependent) functional.
To deduce the answer, we write the potential functional
as a functional of vS (r) rather than v(r)[16]:
Error (a.u.)
0.04
scKX*
0.02
0
2
4
6
(2)
where Ψ0 [v] is the ground-state wavefunction of v(r), so
Z
E0 = min F [˜
v ] + dr n[˜
v ](r) v(r)
(3)
0.06
scX*
(1)
where the search is over all normalized, antisymmetric
ˆ
Ψ, and Tˆ is the kinetic energy
Poperator, Vee the electronˆ
electron repulsion, and V = i v(ri ) the one-body operator. We use Hartree atomic units (e2 = ~ = me = 1)
and suppress spin indices for simplicity. The universal
potential functional[17] is
0.08
LDAX
Total pages: 6
8
N
FIG. 1. Error in the total energy made by LDA exchange
(LDAX), semiclassical exchange (scX), and semiclassical kinetic and exchange (scKX) for N spin-unpolarized, interacting fermions in a well (see Tab. I).
F¯ [vS ] = F [v[vS ]] = TS [vS ] + U [vS ] + EXC [vS ],
Here we go beyond the non-interacting case by including both Hartree and exchange components of the
electron-electron interaction to illustrate the promise of
PFT for avoiding the cost of orbital-dependent DFT calculations. In Fig. 1, we show the errors made in the total
energy of one-dimensional electrons in a potential with
box boundaries, using a recently developed semiclassical
PFT approximation[22]. scX is a simple explicit formula
for the exchange energy in terms of the KS potential,
and here is evaluated on the self-consistent potential in
a LDA exchange (LDAX) calculation. Even for only one
occupied orbital, the error is less than 5% of LDAX, and
is negligible for two or more orbitals, even though the exchange energy grows. If such a formula existed for three
dimensions, the cost of (almost) EXX would be vanishingly small. We also (i) develop the KS equation of PFT
for interacting particles without recourse to DFT quanti-
(4)
i.e., all are functionals of the KS potential (which is
uniquely determined by v(r)), where U is the Hartree
energy and EXC is everything else. As mentioned above,
with a given nS [vS ](r), we can determine TS and U . Applying Eq. (3), but now searching over trial KS potentials,
yields, via the Hohenberg-Kohn theorem[14]
Z
¯
E0 = min F [vS ] + dr nS [vS ](r) v(r)
(5)
vS
where we call the minimizing KS potential v˜S (r). To find
v˜S (r), we use the Euler equation[16]:
δEv0 [vS ] =0
(6)
δvS (r) v˜S
2
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−5 sin2 (πx), and repelling each other via exp(−αu) with
α = 4. These parameters are chosen so that even for
N = 2, the condition on the Fermi energy is satisfied.
We first define what exact calculation we shall use to
analyze our results. In this context, it is a full OEP calculation using the exact orbital expression for exchange.
Such a calculation produces the exact KS kinetic and exchange energies and KS potential on the self-consistent
EXX density for the problem. Next, we define LDAX and
check its performance. The LDAX energy per electron is
for both the interacting and non-interacting systems, and
equate potentials:
Z
δEHXC [vS ] 0
0 −1
0
,
vS [˜
vS ](r) = v0 (r) + dr χS [˜
vS ](r , r)
δvS (r0 ) v˜S
(7)
where EHXC
=
U + EXC , χS [˜
vS ](r0 , r)
=
δnS [vS ](r0 )/δvS (r)|v˜S is the one-body density-density
response function and:
Z
δTS [vS ] 0
−1
vS [˜
vS ](r) = − χS [˜
vS ]
.
(8)
δvS v˜S
LDA
(n(x)) = −
X
as shown in Ref. 16. Note that vS0 = v˜S only if TS [vS ] and
n[vS ] together satisfy the noninteracting Euler equation,
such as for the exact functionals, as in Ref. [14]. The
solution of Eq. (7) yields the minimizing KS potential
v˜S (r), once nS [vS ](r) and EHXC [vS ] are given. But since
Eq. (7) requires computing the inverse of χS , which becomes costly with increasing particle number, we instead
directly minimize Eq. (5).
We next turn to actual calculations, using approximate
potential functionals. Applying similar integration techniques in the complex energy plane as in Refs. 15 and 18,
we obtain a semiclassical potential functional approximation (PFA) to the one-body reduced density matrix
X λ sin[θλ (x, x0 )]cosec[αλ (x, x0 )/2]
F
F
p
,
0)
2T
k
(x)k
(x
F
F
F
λ=+,−
(9)
of N fermions in a one-dimensional potential, inside a
box, whose chemical potential is above the potential everywhere. Here θ± (x,R x0 ) = θ(x) ± θ(x0 ), α± (x, x0 ) =
x
α(x) ± α(x0 ), θ(x) = 0 dx0 k(x0 ) denotes the semiclasp
sical phase, k(x) = 2(E − v(x)) the Rwave vector, E is
x
the energy, α(x) = πτ (x)/T , τ (x) = 0 dx0 k −1 (x0 ) the
traveling time of a classical particle in the potential v(x)
from one boundary to the point x at a given energy, and
T = τ (L)[15]. A subscript F denotes evaluation at the
Fermi energy, which is found by requiring the wavefunctions to vanish at the edge, i.e., ΘF (L) = (N + 1/2)π.
The derivation and implications for DFT of this expression is given elsewhere[22]. As x → x0 , the diagonal reduces to the known semiclassical approximation for the
density[15] and two derivatives yield the corresponding
approximation to the non-interacting kinetic energy[17].
For a given electron-electron repulsion, vee (u), where
u = |x − x0 | denotes the separation between electrons,
the semiclassical exchange is:
1
EX [vS ] = −
2
Z∞ Z∞
arctan β
ln(1 + β 2 )
+
π
2πβ
(11)
with β = 2πn(x)/α. In Tab. I we report exact total energies and errors of several approximate calculations, as a
function of (double) occupation of orbitals. We see that
LDAX makes a substantial error for N = 2 which grows
with N , although EX itself grows, so the fractional error is
vanishing (as it must[15]) as N → ∞. A modern generalized gradient approximation might reduce this error by a
factor of 2 or 3. In Tab. II, we list the total energy and its
various components for four particles in the well. Since
the energy error is almost entirely given by the exchange
error, this means the LDAX density and component energies are very accurate, and the corresponding LDAX KS
potential quite accurate. Due to the variational principle,
the small differences in the different energy components
almost cancel.
Our first new calculation is a post-LDA calculation
of the exchange energy using the semiclassical approximation of Eq. 10, i.e., EXsc [vSLDAX ]. This is orbital-free
exchange but using the potential rather than the density
as the basic variable. The error is plotted in Fig. 1, and
tabulated next to the LDAX results in Tabs. I and II, denoted scX*, where the * indicates a non-variational calculation. Even for N = 2, the error is an order of magnitude
smaller than LDAX. As N grows, the error shrinks very
rapidly, even in absolute terms, because the semiclassical corrections to LDAX capture the leading corrections
in powers of 1/N [18, 19]. In the next column over, we
even use the semiclassical kinetic energy as well (scKX)
on the LDAX KS potential, and see that, although the
errors can be much larger, they are still far below those
of LDAX. These results show that the semiclassical exchange and even kinetic energy can be extracted from a
simple LDAX self-consistent calculation, yielding much
smaller errors than LDAX. Thus results almost identical
to expensive EXX OEP calculations can be found at essentially no cost with a PFA exchange that includes the
leading asymptotic corrections to LDAX.
But such a recipe, while showing the accuracy of resulting exchange energies quickly, can be criticized for
not being variational, i.e., not the result of any selfconsistent minimization. Our second type of calculation
is to again use the semiclassical PFT exchange witihn
a regular KS-DFT calculation. The resulting expression for the total energy is then minimized. We expand the KS potential in Chebyshev polynomials and
γSsc (x, x0 ) =
sc
Total pages: 6
dx dx0 |γSsc [vS ](x, x0 )|2 vee (u). (10)
−∞ −∞
We next test both the accuracy and the stability of the
semiclassical approximations relative to standard DFT.
In all cases, we put the ‘electrons’ in pairs in a 1d
box of unit length, with a one-body potential v(x) =
3
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N
2
4
6
8
E EXX
2.81
39.04
126.10
283.70
EXEXX
−0.52
−1.26
−2.10
−2.98
LDAX scX*
41.72 −1.79
58.41 −0.15
70.24
0.14
77.91
0.08
error (a.u.) EX energy density (a.u.)
TABLE I. Total EXX energy and respective errors of selfconsistent as well as perturbative post-LDAX(*) calculations within LDAX, scX, and scKX for N spin-unpolarized
fermions interacting via exp(−4u) in an external potential
v(x) = −5 sin2 (πx) within a box of unit length.
error·103
scKX* scX
scKX
1.40 −3.10 −29.60
5.89 −3.86 −1.14
0.53 −1.20
0.47
−0.40 −0.10 −1.76
E
TS
Vext
U
EX
39.04
49.44
−12.72
3.58
−1.26
LDAX
58.41
1.22
−1.38
0.003
58.56
error·103
scX
−3.86
0.34
0.07
0.02
−4.29
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
0.06
0.04
0.02
0
-0.02
EXX
-0.4
TABLE II. Energy components of self-consistent calculations
within LDAX, semiclassical exchange (scX), and a semiclassical approximation of all energy components (scKX) for 4
’electrons’ in the same problem as in Tab. I.
EXX
Total pages: 6
LDAX
-0.2
scX
0
x (a.u.)
scKX
0.2
0.4
FIG. 2. Exchange energy density of 4 spin-unpolarized
fermions for the same problem as in Tab. I. The upper plot
shows the EXX energy as well as result from a self-consistent
calculation via LDAX, scX, and scKX. The respective errors
are plotted in the lower panel.
scKX
−1.14
1.22
4.56
−5.90
−1.02
Fig. 2, and their errors. The scX density greatly improves
over the LDAX density everywhere in space (except
where LDAX accidentally matches the exact value). This
is in stark contrast to the well-known difficulty of defining
and comparing energy densities in generalized gradient
approximations and other DFT approximations[27].
Finally, our piece de resistance is to run a pure PFT
calculation, using semiclassical expressions for all energy
components, not just the exchange energy, by directly
minimizing Eq. (5). This is a true orbital-free calculation,
the PFT analog of orbital-free DFT, and we compare its
results to a full OEP EXX calculation. We denote this
scKX, and its results are in the far right columns of Tabs.
I and II.
First, note that because we have now approximated
the kinetic energy, we would be doing extremely well to
even match an LDAX calculation. However, we see that
in every case, the errors are smaller than LDAX. This is
the basic criterion for a successful orbital-free functional:
its errors are smaller than typical errors in XC approximations. However, we also note that for any N > 2, its
errors are so small (below 2 mH) that they match those
of exact exchange for most practical purposes. Finally,
note that inaccuracies for N = 1 or 2 do not matter, since
the exchange energy for those cases is known exactly via
the Hartree energy.
Looking more closely, it is remarkable that scKX is
more accurate than scX for N = 2 and 4. If we look at
the individual energy components in Tab. II, we see that,
e.g., the Hartree energy is far more accurate in scX than
scKX, while the reverse is true for EX . This implies that
the density is quite inaccurate in scKX, but substantial
cancellation of errors occurs. To see this, in Fig. 3 we
plot both the KS potentials and density errors for the
different calculations, showing the much greater errors
use the Nelder-Mead method[23, 24] to optimize the expansion coefficients. A similar technique was used for
the EXX case[25, 26], where the exchange energy was
the usual Fock integral. We should point out treating
this method variationally required additional constraints
than the perturbative case. For certain systems the minimization would find pathological potentials that behave
badly near the box boundaries but nevertheless minimize
the total energy. Conveniently the semiclassical approximations developed contain an error check in the form
of the normalization of the semiclassical density. If this
normalization deviated by 1% or more from N , we add
a large penalty to the total energy. This is then used to
exclude such potentials that lie far from the domain of
applicability of our approximations and leads to the good
results of Tab. I.
In Tabs. I and II, next to the scKX* columns, we list
the scX results of this procedure. The error remains
much smaller than that of LDAX, and rapidly reduces
with increasing N . This is consistent with our previous semiclassical approximations for the density and kinetic energy[15, 17–19]. However, errors are also typically
much larger than those of the non-self-consistent calculation (scX*), showing that the variational properties are
less robust than in LDAX. This is not surprising, given
that LDAX satisfies a crucial symmetry condition that
scX does not[17, 19]. This is related to the very incorrect
local minima that the procedure finds if not restrained,
as mentioned above.
To illusrate better the improvement in going from
LDAX to scX, we plot the exchange energy densities in
4
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[1] P. Hohenberg and W. Kohn, “Inhomogeneous electron
gas,” Phys. Rev. 136, B864–B871 (1964).
[2] W. Kohn and L. J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev.
140, A1133–A1138 (1965).
[3] A.D. Becke, “Density-functional exchange-energy approximation with correct asymptotic behavior,” Phys.
Rev. A 38, 3098–3100 (1988).
[4] John P. Perdew, Kieron Burke, and Matthias Ernzerhof, “Generalized gradient approximation made simple,”
Phys. Rev. Lett. 77, 3865–3868 (1996), ibid. 78, 1396(E)
(1997).
[5] A.D. Becke, “Density-functional thermochemistry. iii. the
role of exact exchange,” J. Chem. Phys. 98, 5648–5652
(1993).
[6] Kieron Burke, “Perspective on density functional the-
ory,” The Journal of Chemical Physics 136, 150901
(2012).
Weitao Yang and Qin Wu, “Direct method for optimized
effective potentials in density-functional theory,” Phys.
Rev. Lett. 89, 143002 (2002).
Stephan K¨
ummel and Leeor Kronik, “Orbital-dependent
density functionals: Theory and applications,” Rev.
Mod. Phys. 80, 3–60 (2008).
Henk Eshuis and Filipp Furche, “A parameter-free density functional that works for noncovalent interactions,”
The Journal of Physical Chemistry Letters 2, 983–989
(2011), http://pubs.acs.org/doi/pdf/10.1021/jz200238f.
Robert A. Donnelly and Robert G. Parr, “Elementary
properties of an energy functional of the firstorder reduced density matrix,” The Journal of Chemical Physics
69, 4431–4439 (1978).
KS potential (a.u.)
Thus minimizing our PFA reproduces the result of
a self-consistent EXX KS calculation. Furthermore, as
the number of electrons increases, not only does the
PFA computational effort not increase significantly, but
the accuracy also increases. The Fock integral required
in EXX or hybrid calculations scales formally as Ω2 N 2
where Ω is the number of real space grid points used in
our 1d box. Our semiclassical expression simply scales
as Ω2 . As N increases, Ω should scale linearly in order to preserve the ratio of grid points to orbital nodes.
Thus the Fock integral scales as N 4 while our approximation scales much more favourably as N 2 . In quantum
chemistry, evaluation of the Fock energy has been the
focus of much effort to improve the scaling, but at best
the scaling can be reduced to roughly N 3 (e.g., when
localized basis sets and various optimization techniques
are used). Thus our scaling remains advantageous. The
scKX calculation is completely orbital-free and so avoids
solving the KS equation. Either due to direct diagonalization or the orthogonalization of orbitals depending on
the method used, the KS scheme scales as N 3 , while
scKX scales as N 2 due to exchange (the other energy
components scale as N ). Thus the PFT method can effectively reproduce the result of an EXX KS calculation
while requiring a fraction of the computational cost. Substituting EXX with our semiclassical exchange may also
be done in the hybrid functional approach to DFT (although treated within the OEP framework), where the
fraction of EXX mixed in with a standard DFT functional may be replaced. Calculating this EXX energy is
often the costliest part for hybrid calculations. By using
the semiclassical PFA exchange one could vastly speed up
such calculations without a significant loss in accuracy.
In conclusion we have shown that an approximation to
the exchange energy is almost exact and does not require
any orbital information in the framework of PFT. In both
accuracy and efficiency, the PFT method performs better
than high-level KS-DFT calculations. If ongoing work to
extend the method to 3d systems is successful, electronic
structure calculations could be sped up by serveral orders
of magnitudes, allowing large systems that are currently
out of reach with density functional methods to be studied.
PE, SP, and EKUG acknowledge funding by the European Commission (Grant No. FP7-NMP-CRONOS).
EKUG thanks the KITP at UCSB for splendid hospitality. This research was supported in part by the National
Science Foundation under Grant No. NSF PHY11-25915.
KB and AC acknowledge support by National Science
Foundation under Grant No. CHE-1112442 NSF.
0
-2
-4
-6
∆ n(x) (a.u.)
-8
EXX
LDAX
scX
scKX
0.02
0
-0.02
-0.4
-0.2
0
x (a.u.)
0.2
0.4
FIG. 3. Upper plot: Converged KS potentials of EXX, LDAX,
scX, and scKX runs for the same problem as in Tab. I with
4 spin-unpolarized fermions. Lower plot: Error in the respective, converged densities with respect to EXX.
in scKX. However, the cancellation of errors might well
be due to the balanced nature of the calculation, since
all energy components have been derived from a single
approximation for the density matrix[16]. Only extensive
testing for many different circumstances can determine if
this is a general phenomenon and if so, where it fails.
[7]
[8]
[9]
[10]
5
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[11] Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernzerhof, “Hybrid functionals based on a screened coulomb
potential,” The Journal of Chemical Physics 118, 8207–
8215 (2003).
[12] Jochen Heyd, Juan E. Peralta, Gustavo E. Scuseria, and
Richard L. Martin, “Energy band gaps and lattice parameters evaluated with the heyd-scuseria-ernzerhof screened
hybrid functional,” The Journal of Chemical Physics
123, 174101 (2005).
[13] E. Bylaska, K. Tsemekhman, N. Govind, and M. Valiev,
“Large-scale plane-wave-based density functional theory:
Formalism, parallelization, and applications,” in Computational Methods for Large Systems: Electronic Structure Approaches for Biotechnology and Nanotechnology,
edited by J. R. Reimers (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2011).
[14] Weitao Yang, Paul W. Ayers, and Qin Wu, “Potential functionals: Dual to density functionals and solution
to the v-representability problem,” Phys. Rev. Lett. 92,
146404 (2004).
[15] Peter Elliott, Donghyung Lee, Attila Cangi, and Kieron
Burke, “Semiclassical origins of density functionals,”
Phys. Rev. Lett. 100, 256406 (2008).
[16] E K U Gross and C R Proetto, “Adiabatic connection and
the kohn-sham variety of potential-functional theory,” J.
Chem. Theory Comput. 5, 844–849 (2009).
[17] Attila Cangi, Donghyung Lee, Peter Elliott, Kieron
Burke, and E. K. U. Gross, “Electronic structure via
potential functional approximations,” Phys. Rev. Lett.
106, 236404 (2011).
[18] Attila Cangi, Donghyung Lee, Peter Elliott, and Kieron
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
6
Total pages: 6
Burke, “Leading corrections to local approximations,”
Phys. Rev. B 81, 235128 (2010).
Attila Cangi, E. K. U. Gross, and Kieron Burke, “Potential functionals versus density functionals,” Phys. Rev. A
88, 062505 (2013).
J. Schwinger, “Thomas-fermi model: The leading correction,” Phys. Rev. A 22, 1827 (1980).
Julian Schwinger, “Thomas-fermi model: The second
correction,” Phys. Rev. A 24, 2353–2361 (1981).
Attila Cangi, Peter Elliott, E. K. U. Gross, and Kieron
Burke, in prep. (2014).
J. A. Nelder and R. Mead, “A simplex method for function minimization,” The Computer Journal 7, 308–313
(1965).
W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, “Numerical recipes,” (Cambridge University Press,
1992) Chap. Subroutine amoeba.
Francois Colonna and Andreas Savin, “Correlation energies for some two- and four-electron systems along the
adiabatic connection in density functional theory,” J.
Chem. Phys. 119, 2828–2835 (1999).
Degao Peng, Bo Zhao, Aron J. Cohen, Xiangqian Hu,
and Weitao Yang, “Optimized effective potential for calculations with orbital-free potential functionals,” Molecular Physics 110, 925–934 (2012).
John P. Perdew, Adrienn Ruzsinszky, Jianwei Sun, and
Kieron Burke, “Gedanken densities and exact constraints
in density functional theory,” The Journal of Chemical
Physics 140, 18 (2014).

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